Dimensionality Testing, Shape-Constrained Neural Network Estimation, and Equilibrium Approximation in Team Games

Abstract

Modern advancements in computational technology have catalyzed an explosion in data dimensionality, enabling researchers to model intricate relationships among multitudinous variables simultaneously; however, these advancements also challenge the application of many useful classical theories (for instance, the $F$-test which is one of the most widely used tests fails with increasing dimensions alongside sample growth). Although some modifications of classical theories, such as the corrected $F$-test, have been proposed to mitigate the high-dimensional bias, a more fundamental question about whether the dimension is ``high" has not been addressed. In Chapter 1, we tackle this problem by proposing a test statistic, which is asymptotically normal, to distinguish high from non-high dimensionality for vectors. In large sample sizes, our test has the correct size when the dimension is ``high"; and it is asymptotically powerful against all alternatives with ``low" dimensions. A set of simulation studies demonstrates that our test achieves satisfactory size and power even in small to moderate sample sizes.

Deep neural networks (DNNs), as the most modern advancement, have demonstrated exceptional performance across a wide range of domains, such as image classification \citep{NIPS2012_c399862d,he2016deep}, speech recognition \citep{hinton2012deep}, natural language processing \citep{mikolov2013efficient,vaswani2017attention} and game intelligence \citep{silver2016mastering,perolat2022mastering}. DNNs can be conceptualized as a generalization of classical sieve estimators, transitioning from ``single-layer" to ``multi-layer" architectures. This architectural evolution substantially expands the model's parameter space, but diminishing the interpretability of parameters and the model itself. In Chapter 2, we provide the consistency and asymptotic normality of the DNN-based estimator for point estimation and extend these properties the case with prior knowledge of shapes.

The incorporation of prior knowledge into neural networks to enhance model performance has evolved into a fundamental methodology, being widely used in real-world applications. The abstract prior knowledge, particularly the shape restrictions such as monotonicity, convexity, and homogeneity, hold critical importance within economic modeling frameworks, where functional specifications like production and utility functions inherently require certain shape restrictions to ensure both mathematical and theoretical validity in economic analyses. However, current methods exhibit fundamental limitations in effectively embedding abstract knowledge, since those methods either implement loss function modifications that fail to guarantee the desired shape, or require specifically designed structures through restricting the number or type of layers to certain forms--a parametric imposition that essentially limits model expressiveness. Our shape-restricted approximator is shown to be consistent and asymptotically normal under mild regularity conditions. Based on case studies, the proposed method outperforms the existing shape-restricted networks.

Two-team zero-sum games model a broad range of real-world competitive interactions more effectively than traditional two-player games, yet they remain relatively underexplored. Existing successful approaches for two-player games, including Go, poker, and Stratego, are difficult to extend directly to team settings, as they cannot simultaneously account for inter-team competition and intra-team cooperation. In chapter 3, we investigate two-team zero-sum games under the setting where teammates cannot communicate, with the goal of approximating the team Nash equilibrium. Unlike existing methods such as TMECor \citep{TMECor_Team-PSRO} and GFXP \citep{FXP_team_comp}, which rely on communication or correlation devices by treating each team as a centralized joint player, our method directly tackles decentralized team settings without such coordination mechanisms. We propose a novel extension of Policy-Space Response Oracles (PSRO) \citep{PSRO} that incorporates multi-agent reinforcement learning (MARL) to compute joint cooperative best responses during population expansion. By leveraging shared team rewards, the proposed method captures cooperation within teams while preserving competitive optimization across teams. The proposed algorithm demonstrates a stronger tendency to converge to global Nash equilibria compared with classical PSRO based on some experimental results.